In our proof, at no point do we need to approximate a continuous map by smooth maps.
Instead, the proof relies on a result about Čech nerves of open coverings from [1]}, a version of the Nerve Theorem, see e.g. [2]}, [3]}, [4]}, [5]} (concretely, we use [6]}, which is sometimes also called Lurie's Seifert van Kampen Theorem), and a modified two-sided simplicial bar construction for simplicial presheaves, which we introduce in Appendix .
This illustrates that the model categories \({H}_\infty ^{p/i\, I}\)  are of interest beyond their abstract properties: they provide useful tools for doing smooth homotopy theory.
